Exhibit 5.1

May 21, 2020

Hudson Executive Investment Corp.

570 Lexington Avenue, 35^th Floor

New York, NY 10022

Re: Form S-1 Registration Statement

Ladies and Gentlemen:

We have acted as special counsel to Hudson Executive Investment Corp., a Delaware corporation (the “Company”), in connection with the preparation of the Company’s registration statement on Form S-1 initially filed with the U.S. Securities and Exchange Commission (the “Commission”) on May 21, 2020 (the “Registration Statement”), under the Securities Act of 1933, as amended (the “Securities Act”).

The Registration Statement relates to the registration of the offer and sale of up to 34,500,000 Units of the Company (the “Units”) (including up to 4,500,000 Units subject to the Underwriters’ (as defined below) over-allotment option), with each Unit consisting of (i) one share of the Company’s Class A common stock, par value $0.0001 per share (the “Common Stock” and the Common Stock underlying the Units, the “Shares”), for an aggregate of up to 34,500,000 Shares (including up to 4,500,000 Shares included in the Units subject to the Underwriters’ over-allotment option) and (ii) one-half of one redeemable warrant (a “Warrant”), with each whole Warrant entitling the holder to purchase one share of Common Stock, for an aggregate of up to 17,250,000 Warrants (including up to 2,250,000 Warrants included in the Units subject to the Underwriters’ over-allotment option) to be issued under a warrant agreement (the “Warrant Agreement”) to be entered into by the Company and Continental Stock Transfer & Trust Company, as Warrant Agent, pursuant to the terms of an underwriting agreement (the “Underwriting Agreement”) to be executed by the Company and Citigroup Global Markets Inc. and J.P. Morgan Securities LLC, as representatives of the underwriters named therein (the “Underwriters”).

This opinion letter is being furnished in accordance with the requirements of Item 601(b)(5) of Regulation S-K promulgated under the Securities Act.

In rendering the opinions set forth below, we examined and relied upon such certificates, corporate records, agreements, instruments and other documents, and examined such matters of law, that we considered necessary or appropriate as a basis for the opinions. In rendering the opinions set forth below, we have examined and are familiar with originals or copies, certified or otherwise identified to our satisfaction, of (i) the Certificate of Incorporation of the Company, as in effect on the date hereof, (ii) the Bylaws of the Company, as in effect on the date hereof, (iii) the Registration Statement, (iv) the Underwriting Agreement, (v) resolutions of the Board of Directors of the Company relating to, among other matters, the issuance of the Units, the Shares and the Warrants and the filing of the Registration Statement and (vi) such other documents as we have deemed necessary or appropriate as a basis for the opinions set forth below. In our examination, we have assumed the legal capacity of all natural persons, the genuineness of all

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signatures, the authenticity of all documents submitted to us as originals, the conformity to original documents of all documents submitted to us as certified or photostatic copies, the authenticity of the originals of such latter documents, that all parties to such documents (other than the Company) had the power, corporate or other, to enter into and perform all obligations thereunder and all such documents have been duly authorized by all requisite action, corporate or other, and duly executed and delivered by all parties thereto (other than the Company). As to any facts material to the opinions expressed herein that we did not independently establish or verify, we have relied upon oral or written statements and representations of officers and other representatives of the Company and others. In rendering the opinions set forth below, we have further assumed that, before the issuance of the Units, the Shares and the Warrants, (i) the Registration Statement will have become effective under the Securities Act and (ii) the conditions to consummating the transactions contemplated by the Underwriting Agreement will have been satisfied or duly waived and such transactions are consummated.

Based upon the foregoing and subject to the assumptions, qualifications and limitations set forth herein, we are of the opinion that:

1. When the Units are delivered to the Underwriters against payment of the agreed consideration therefor in accordance with the Underwriting Agreement, each Unit will be a valid and binding agreement of the Company, enforceable against the Company in accordance with its terms.

2. Each Share included in the Units has been duly authorized and, when the Units are delivered to the Underwriters against payment of the agreed consideration therefor in accordance with the Underwriting Agreement, each Share included in the Units will be validly issued, fully paid and non-assessable.

3. When the Units are delivered to the Underwriters against payment of the agreed consideration therefor in accordance with the Underwriting Agreement, each Warrant included in the Units will be a valid and binding agreement of the Company, enforceable against the Company in accordance with its terms.

The opinions expressed herein are based upon and limited to the laws of the State of New York and the General Corporation Law of the State of Delaware, including the statutory provisions, all applicable provisions of the Delaware Constitution and reported judicial decisions interpreting the foregoing. We express no opinion herein as to any other laws, statutes, regulations or ordinances. The opinions expressed herein that are based on the laws of the State of New York are limited to the laws generally applicable in transactions of the type covered by the Registration Statement.

We hereby consent to the filing of this opinion letter as Exhibit 5.1 to the Registration Statement and to the reference to our firm under the caption “Legal Matters” in the prospectus included in the Registration Statement. In giving such consent, we do not thereby admit that we are experts within the meaning of the Securities Act or the rules and regulations of the Commission or that this consent is required by Section 7 of the Securities Act.

Very truly yours,

/s/ Winston & Strawn LLP